fz1isolem - Metamath Proof Explorer

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Theorem fz1isolem 14427

Description: Lemma for fz1iso 14428. (Contributed by Mario Carneiro, 2-Apr-2014.)

**Hypotheses**
Ref	Expression
fz1iso.1	⊢ 𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
fz1iso.2	⊢ 𝐵 = (ℕ ∩ (^◡ < “ {((♯‘𝐴) + 1)}))
fz1iso.3	⊢ 𝐶 = (ω ∩ (^◡𝐺‘((♯‘𝐴) + 1)))
fz1iso.4	⊢ 𝑂 = OrdIso(𝑅, 𝐴)

**Assertion**
Ref	Expression
fz1isolem	⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴))

Distinct variable groups: 𝑓,𝑛,𝐴 𝐵,𝑓 𝑓,𝐺 𝑓,𝑂 𝑅,𝑓 Allowed substitution hints: 𝐵(𝑛) 𝐶(𝑓,𝑛) 𝑅(𝑛) 𝐺(𝑛) 𝑂(𝑛)

**Proof of Theorem fz1isolem**
Step	Hyp	Ref	Expression
1		hashcl 14321	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
2	1	adantl 481	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (♯‘𝐴) ∈ ℕ₀)
3		nnuz 12870	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
4		1z 12597	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℤ
5		fz1iso.1	. . . . . . . . . . . . 13 ⊢ 𝐺 = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 1) ↾ ω)
6	4, 5	om2uzisoi 13924	. . . . . . . . . . . 12 ⊢ 𝐺 Isom E , < (ω, (ℤ_≥‘1))
7		isoeq5 7321	. . . . . . . . . . . 12 ⊢ (ℕ = (ℤ_≥‘1) → (𝐺 Isom E , < (ω, ℕ) ↔ 𝐺 Isom E , < (ω, (ℤ_≥‘1))))
8	6, 7	mpbiri 258	. . . . . . . . . . 11 ⊢ (ℕ = (ℤ_≥‘1) → 𝐺 Isom E , < (ω, ℕ))
9	3, 8	ax-mp 5	. . . . . . . . . 10 ⊢ 𝐺 Isom E , < (ω, ℕ)
10		isocnv 7330	. . . . . . . . . 10 ⊢ (𝐺 Isom E , < (ω, ℕ) → ^◡𝐺 Isom < , E (ℕ, ω))
11	9, 10	ax-mp 5	. . . . . . . . 9 ⊢ ^◡𝐺 Isom < , E (ℕ, ω)
12		nn0p1nn 12516	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ₀ → ((♯‘𝐴) + 1) ∈ ℕ)
13		fz1iso.2	. . . . . . . . . 10 ⊢ 𝐵 = (ℕ ∩ (^◡ < “ {((♯‘𝐴) + 1)}))
14		fz1iso.3	. . . . . . . . . . 11 ⊢ 𝐶 = (ω ∩ (^◡𝐺‘((♯‘𝐴) + 1)))
15		fvex 6904	. . . . . . . . . . . . 13 ⊢ (^◡𝐺‘((♯‘𝐴) + 1)) ∈ V
16	15	epini 6095	. . . . . . . . . . . 12 ⊢ (^◡ E “ {(^◡𝐺‘((♯‘𝐴) + 1))}) = (^◡𝐺‘((♯‘𝐴) + 1))
17	16	ineq2i 4209	. . . . . . . . . . 11 ⊢ (ω ∩ (^◡ E “ {(^◡𝐺‘((♯‘𝐴) + 1))})) = (ω ∩ (^◡𝐺‘((♯‘𝐴) + 1)))
18	14, 17	eqtr4i 2762	. . . . . . . . . 10 ⊢ 𝐶 = (ω ∩ (^◡ E “ {(^◡𝐺‘((♯‘𝐴) + 1))}))
19	13, 18	isoini2 7339	. . . . . . . . 9 ⊢ ((^◡𝐺 Isom < , E (ℕ, ω) ∧ ((♯‘𝐴) + 1) ∈ ℕ) → (^◡𝐺 ↾ 𝐵) Isom < , E (𝐵, 𝐶))
20	11, 12, 19	sylancr 586	. . . . . . . 8 ⊢ ((♯‘𝐴) ∈ ℕ₀ → (^◡𝐺 ↾ 𝐵) Isom < , E (𝐵, 𝐶))
21	2, 20	syl 17	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺 ↾ 𝐵) Isom < , E (𝐵, 𝐶))
22		nnz 12584	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ ℕ → 𝑓 ∈ ℤ)
23	2	nn0zd 12589	. . . . . . . . . . . . 13 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (♯‘𝐴) ∈ ℤ)
24		eluz 12841	. . . . . . . . . . . . 13 ⊢ ((𝑓 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → ((♯‘𝐴) ∈ (ℤ_≥‘𝑓) ↔ 𝑓 ≤ (♯‘𝐴)))
25	22, 23, 24	syl2anr 596	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑓 ∈ ℕ) → ((♯‘𝐴) ∈ (ℤ_≥‘𝑓) ↔ 𝑓 ≤ (♯‘𝐴)))
26		zleltp1 12618	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (𝑓 ≤ (♯‘𝐴) ↔ 𝑓 < ((♯‘𝐴) + 1)))
27	22, 23, 26	syl2anr 596	. . . . . . . . . . . . 13 ⊢ (((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑓 ∈ ℕ) → (𝑓 ≤ (♯‘𝐴) ↔ 𝑓 < ((♯‘𝐴) + 1)))
28		ovex 7445	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝐴) + 1) ∈ V
29		vex 3477	. . . . . . . . . . . . . . 15 ⊢ 𝑓 ∈ V
30	29	eliniseg 6093	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝐴) + 1) ∈ V → (𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)}) ↔ 𝑓 < ((♯‘𝐴) + 1)))
31	28, 30	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)}) ↔ 𝑓 < ((♯‘𝐴) + 1))
32	27, 31	bitr4di 289	. . . . . . . . . . . 12 ⊢ (((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑓 ∈ ℕ) → (𝑓 ≤ (♯‘𝐴) ↔ 𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)})))
33	25, 32	bitr2d 280	. . . . . . . . . . 11 ⊢ (((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) ∧ 𝑓 ∈ ℕ) → (𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)}) ↔ (♯‘𝐴) ∈ (ℤ_≥‘𝑓)))
34	33	pm5.32da 578	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((𝑓 ∈ ℕ ∧ 𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)})) ↔ (𝑓 ∈ ℕ ∧ (♯‘𝐴) ∈ (ℤ_≥‘𝑓))))
35	13	elin2 4197	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐵 ↔ (𝑓 ∈ ℕ ∧ 𝑓 ∈ (^◡ < “ {((♯‘𝐴) + 1)})))
36		elfzuzb 13500	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (1...(♯‘𝐴)) ↔ (𝑓 ∈ (ℤ_≥‘1) ∧ (♯‘𝐴) ∈ (ℤ_≥‘𝑓)))
37		elnnuz 12871	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ ℕ ↔ 𝑓 ∈ (ℤ_≥‘1))
38	37	anbi1i 623	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℕ ∧ (♯‘𝐴) ∈ (ℤ_≥‘𝑓)) ↔ (𝑓 ∈ (ℤ_≥‘1) ∧ (♯‘𝐴) ∈ (ℤ_≥‘𝑓)))
39	36, 38	bitr4i 278	. . . . . . . . . 10 ⊢ (𝑓 ∈ (1...(♯‘𝐴)) ↔ (𝑓 ∈ ℕ ∧ (♯‘𝐴) ∈ (ℤ_≥‘𝑓)))
40	34, 35, 39	3bitr4g 314	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝑓 ∈ 𝐵 ↔ 𝑓 ∈ (1...(♯‘𝐴))))
41	40	eqrdv 2729	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐵 = (1...(♯‘𝐴)))
42		isoeq4 7320	. . . . . . . 8 ⊢ (𝐵 = (1...(♯‘𝐴)) → ((^◡𝐺 ↾ 𝐵) Isom < , E (𝐵, 𝐶) ↔ (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), 𝐶)))
43	41, 42	syl 17	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((^◡𝐺 ↾ 𝐵) Isom < , E (𝐵, 𝐶) ↔ (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), 𝐶)))
44	21, 43	mpbid 231	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), 𝐶))
45		fz1iso.4	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑂 = OrdIso(𝑅, 𝐴)
46	45	oion 9535	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ Fin → dom 𝑂 ∈ On)
47	46	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ∈ On)
48		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
49		wofi 9296	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
50	45	oien 9537	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom 𝑂 ≈ 𝐴)
51	48, 49, 50	syl2anc 583	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ≈ 𝐴)
52		enfii 9193	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ Fin ∧ dom 𝑂 ≈ 𝐴) → dom 𝑂 ∈ Fin)
53	48, 51, 52	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ∈ Fin)
54	47, 53	elind 4194	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ∈ (On ∩ Fin))
55		onfin2 9235	. . . . . . . . . . . . . . 15 ⊢ ω = (On ∩ Fin)
56	54, 55	eleqtrrdi 2843	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ∈ ω)
57		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω) = (rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)
58		0z 12574	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℤ
59	5, 57, 4, 58	uzrdgxfr 13937	. . . . . . . . . . . . . . 15 ⊢ (dom 𝑂 ∈ ω → (𝐺‘dom 𝑂) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + (1 − 0)))
60		1m0e1 12338	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
61	60	oveq2i 7423	. . . . . . . . . . . . . . 15 ⊢ (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + (1 − 0)) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + 1)
62	59, 61	eqtrdi 2787	. . . . . . . . . . . . . 14 ⊢ (dom 𝑂 ∈ ω → (𝐺‘dom 𝑂) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + 1))
63	56, 62	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝐺‘dom 𝑂) = (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + 1))
64	51	ensymd 9005	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ dom 𝑂)
65		cardennn 9982	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ≈ dom 𝑂 ∧ dom 𝑂 ∈ ω) → (card‘𝐴) = dom 𝑂)
66	64, 56, 65	syl2anc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) = dom 𝑂)
67	66	fveq2d 6895	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘(card‘𝐴)) = ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂))
68	57	hashgval 14298	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Fin → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
69	68	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘(card‘𝐴)) = (♯‘𝐴))
70	67, 69	eqtr3d 2773	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) = (♯‘𝐴))
71	70	oveq1d 7427	. . . . . . . . . . . . 13 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (((rec((𝑛 ∈ V ↦ (𝑛 + 1)), 0) ↾ ω)‘dom 𝑂) + 1) = ((♯‘𝐴) + 1))
72	63, 71	eqtrd 2771	. . . . . . . . . . . 12 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝐺‘dom 𝑂) = ((♯‘𝐴) + 1))
73	72	fveq2d 6895	. . . . . . . . . . 11 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺‘(𝐺‘dom 𝑂)) = (^◡𝐺‘((♯‘𝐴) + 1)))
74		isof1o 7323	. . . . . . . . . . . . 13 ⊢ (𝐺 Isom E , < (ω, ℕ) → 𝐺:ω–1-1-onto→ℕ)
75	9, 74	ax-mp 5	. . . . . . . . . . . 12 ⊢ 𝐺:ω–1-1-onto→ℕ
76		f1ocnvfv1 7277	. . . . . . . . . . . 12 ⊢ ((𝐺:ω–1-1-onto→ℕ ∧ dom 𝑂 ∈ ω) → (^◡𝐺‘(𝐺‘dom 𝑂)) = dom 𝑂)
77	75, 56, 76	sylancr 586	. . . . . . . . . . 11 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺‘(𝐺‘dom 𝑂)) = dom 𝑂)
78	73, 77	eqtr3d 2773	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺‘((♯‘𝐴) + 1)) = dom 𝑂)
79	78	ineq2d 4212	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (ω ∩ (^◡𝐺‘((♯‘𝐴) + 1))) = (ω ∩ dom 𝑂))
80		ordom 7869	. . . . . . . . . . 11 ⊢ Ord ω
81		ordelss 6380	. . . . . . . . . . 11 ⊢ ((Ord ω ∧ dom 𝑂 ∈ ω) → dom 𝑂 ⊆ ω)
82	80, 56, 81	sylancr 586	. . . . . . . . . 10 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom 𝑂 ⊆ ω)
83		sseqin2 4215	. . . . . . . . . 10 ⊢ (dom 𝑂 ⊆ ω ↔ (ω ∩ dom 𝑂) = dom 𝑂)
84	82, 83	sylib 217	. . . . . . . . 9 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (ω ∩ dom 𝑂) = dom 𝑂)
85	79, 84	eqtrd 2771	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (ω ∩ (^◡𝐺‘((♯‘𝐴) + 1))) = dom 𝑂)
86	14, 85	eqtrid 2783	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐶 = dom 𝑂)
87		isoeq5 7321	. . . . . . 7 ⊢ (𝐶 = dom 𝑂 → ((^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), 𝐶) ↔ (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), dom 𝑂)))
88	86, 87	syl 17	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ((^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), 𝐶) ↔ (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), dom 𝑂)))
89	44, 88	mpbid 231	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), dom 𝑂))
90	45	oiiso 9536	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
91	48, 49, 90	syl2anc 583	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴))
92		isotr 7336	. . . . 5 ⊢ (((^◡𝐺 ↾ 𝐵) Isom < , E ((1...(♯‘𝐴)), dom 𝑂) ∧ 𝑂 Isom E , 𝑅 (dom 𝑂, 𝐴)) → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)) Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴))
93	89, 91, 92	syl2anc 583	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)) Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴))
94		isof1o 7323	. . . 4 ⊢ ((𝑂 ∘ (^◡𝐺 ↾ 𝐵)) Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴) → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)):(1...(♯‘𝐴))–1-1-onto→𝐴)
95		f1of 6833	. . . 4 ⊢ ((𝑂 ∘ (^◡𝐺 ↾ 𝐵)):(1...(♯‘𝐴))–1-1-onto→𝐴 → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)):(1...(♯‘𝐴))⟶𝐴)
96	93, 94, 95	3syl 18	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)):(1...(♯‘𝐴))⟶𝐴)
97		fzfid 13943	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (1...(♯‘𝐴)) ∈ Fin)
98	96, 97	fexd 7231	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (𝑂 ∘ (^◡𝐺 ↾ 𝐵)) ∈ V)
99		isoeq1 7317	. 2 ⊢ (𝑓 = (𝑂 ∘ (^◡𝐺 ↾ 𝐵)) → (𝑓 Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴) ↔ (𝑂 ∘ (^◡𝐺 ↾ 𝐵)) Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴)))
100	98, 93, 99	spcedv 3588	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom < , 𝑅 ((1...(♯‘𝐴)), 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 Vcvv 3473 ∩ cin 3947 ⊆ wss 3948 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 E cep 5579 Or wor 5587 We wwe 5630 ^◡ccnv 5675 dom cdm 5676 ↾ cres 5678 “ cima 5679 ∘ ccom 5680 Ord word 6363 Oncon0 6364 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 Isom wiso 6544 (class class class)co 7412 ωcom 7859 reccrdg 8413 ≈ cen 8940 Fincfn 8943 OrdIsocoi 9508 cardccrd 9934 0cc0 11114 1c1 11115 + caddc 11117 < clt 11253 ≤ cle 11254 − cmin 11449 ℕcn 12217 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 ...cfz 13489 ♯chash 14295

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-er 8707 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-oi 9509 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-hash 14296

This theorem is referenced by: fz1iso 14428

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